TOOLS FOR BRAMWELL-HOLDSWORTH-PINTON
PROBABILITY DISTRIBUTION
Tiberiu Socaciu
University of Suceava
Suceava, Romania
socaciu@seap.usv.ro
Abstract
This paper is a synthesis of a range of papers presented at various conferences related to distribution
Bramwell-Holdsworth-Pinton. S. T. Bramwell, P. C. W. Holdsworth, J. F. Pinton introduced a new
non-parametric distribution (called BHP) after studying some magnetization problems in 2D.
Probability density function of distribution can be aproximated as a modified GFT (Gumbel-FisherTippitt) distribution.
Keywords
BHP distribution, PP Plot, QQ Plot, Statistical Tests, BET-FI index, Bucharest Exchange
Stock.
1. BHP nonparametric distribution (see [1])
S. T. Bramwell, P. C. W. Holdsworth, J. F. Pinton introduced (see [2]) a new non-parametric
distribution (called BHP) after studying some magnetization problems in 2D. Probability
density function of distribution is:

p ( x) 
dx 1 N 1 1
1 N 1 1 N 1 ix
i
x 1
x2
exp{
ix


[

arctg

ln(
1

)]}



 2 2 N 2 k 1 2k
2 N 2 k 1 2k k 1 2 Nk 2
Nk 4
N 22k

where λk are eigenvalues of adiacency matrix of some specific graphs (see [3]). C. Pennetta,
E. Alfinito, L. Reggiani re-discover (see [4], [5], [6]) this distribution at one specific
electrical rezistivity problem. Structural, a BHP distribution is a particular Gumbel
distribution (see [7]) - named sometime Fischer-Tippett distribution (see [8]). Probability
density function (pdf) of BHP can be aproximated (see [2]) with:
fBHP(x) = K * exp{a * [t - exp(t)]}
where parameters are (see [2], [6]):
t = b * (y - s)
a=π/2
b = 0.936
s = 0.374
K = 2.15
In [3] suggested values are:
t = b * (y - s)
a = 1.5806801
b = 0.9339355
s = 0.3731792
K = 2.1602858
In [5], [6] suggested values are:
t = b * (y - s)
a=π/2
b = 0.936 ± 0.002
s = 0.374 ± 0.001
K = 2.15 ± 0.01
BHP pdf and normal distribution’s pdf in lognormal scale can be see bellow (chart was
generated via a Visual Basic for Application program in Microsoft Excel):
Densitatea BHP si normala
2.8
2.1
1.4
0.7
0.0
-0.7
-1.4
-2.1
-2.8
-3.5
-4.2
-4.9
-5.6
-6.3
-7.0
0.00000
log10(densitate(y))
-2.00000
-4.00000
-6.00000
-8.00000
log10(fBHP(y))
-10.00000
-12.00000
log10(fnorm(y))
y
2. PP Plot and QQ Plot for BHP distribution (see [9])
Let supose that we have a distribution of probability with specific cumulative distribution
function:
F: R[0,1]
Let supose that we have same values of data, denoted
x(1), x(2), ....., x(n).
At first step, after sorting data (from smallest to largest) we obtain the array reordered. A
probability-probability (PP) plot and quantile-quantile (QQ) plot are used to see if a given
set of data follows some specified distribution (identified with his cdf). It should be
approximately linear if the specified distribution is the correct model.
The PP Plot algorithm (adapted from [10]) is:
Algorithm PP_PLOT
Call SORT(x[], n)
For I := 1 To n Do
Call PLOT((I-1/2)/n, F(x(I))
End For
Call LINE((0,0),(1,1))
End Algorithm
where:
PLOT(x,y)
is graphical primitive to put a plot at real coordinates (x,y) and:
LINE(x1,y1,x2,y2)
is graphical primitive to draw a line beetween 2 points gave on real coordinates: (x1, y1) and
(x2, y2).
The QQ Plot Algorithm (adapted from [10]) is:
Algorithm PP_PLOT
Call SORT(x[], n)
For I := 1 To n Do
Call PLOT(x(I),F-1((I-1/2)/n)
End For
Call LINE((x(1),x(1)),(x(n),x(n)))
End Algorithm
Before building a tool for PP Plot and QQ Plot, we need values of cdf of BHP distribution.
We can obtain this from definition:
x
FBHP ( x) 
f
BHP
(t )dt

After studying values of pdf and cdf of BHP distribution, we can rewrite
x
FBHP ( x) 
f
BHP
(t )dt
7
For tabelation of F with a step s, we can use a Darboux sum with an algorithm like:
Algorithm TabelationCdfBHP
Last := 0
For x := -7 To 5 Step s
Last := Last + s * f(x)
F(x) := Last
EndAlgorithm
The inverse function of cdf can be computed easy via a binary search of the argument in
ordered array of cdf tabelation.
3. Statistical test for BHP distribution (see [10])
Lemma: If F iid BHP, then Z(F) iid U(0,1), where:
Z(F) = s - ln (- ln (F)) / b.
Proof: is based on link beetween GFT distribution and uniform distribution and observation
that graphical representation of BHP’s pdf is a symetric of omolog GFT’s pdf graphical
representation.
Note: Uniformity of Z(F) can be verified with a Kolmogorov / Kolmogorov-Smirnov test.
Based on Lemma, and Kolmogorov test, we can build an algorithm for our derived BHP
test:
Algorithm BHP_Test (data[], N, alpha)
Private Z[], s, b
b = 0.9339355
s = 0.3731792
For i = 1 to n do
Z[i] = s - ln (- ln (data[i])) / b
Next i
BHP_Test = KOLMOGOROV(Z[], n, alpha)
End Algorithm
4. Universality hypothesis testing for BET-FI index (see [1])
In [12], Gonçalves and Pinto have proposed a new way to check the universality hypothesis
about stock indexes. They were tested in [12] hypothesis on the component indices Dow
Jones (DJIA30) and Standard & Poors 100 (S & P100) on the New York Stock Exchange.
We present the construction of Gonçalves and Pinto: Let I an index from a Stock Exchange
with composition
COMP(I) = {s1, s2, ..., sn}
where s1, s2, ..., sn are n traded symbols. Let denote P(I,t) closing value on day t of index I
and P(s,t) closing value on day t of symbol s. For a symbol or an index s, we denote daily
return on day t as:
R(s,t) = (P(s,t) - P(s,t-1)) / P(s,t-1)
or alternative, we can use for daily return formula:
R(s,t) = ln P(s,t) – ln P(s,t-1).
For each day t define mean of index at closing the day t as:
m(I,t) = [ Pa(s1,t) + Pa(s2,t) + ... + Pa(sn,t) ] / n,
and dispersion of the day t:
s(I,t) = { [ P2a(s1,t) + P2a(s2,t) + ... + P2a(sn,t) ] / n - m2(I, t) } 1/2.
With these notations define the α-daily fluctuation of the index I as:
df(I,t) = (Pa(I,t) - m(I,t)) / s(I,t).
Based on notations can issue the following conjecture (reviewed in [12] for the U.S. stock
market indices remember where):
Conjecture (of BHP universality): For a = 2/3, the series (df(I,t))t>0 verifies the BHP
distribution.
Note: The type of P (.,t)2/3 are called in [12] as Cubic Root of the Daily Return Squared
(abbreviated CRDRS).
We try to check conjecture with the following values of the Bucharest Stock Exchange (see
[13]), where trading Romanian regional Financial Investment Companies (SIF Moldova,
SIF Muntenia, SIF Transilvania, SIF Oltenia, SIF Banat-Crisana) and is computed daily
BET-FI index. Composition of BET-FI index is:
COMP(BET-FI) = {SIF1, SIF2, SIF3, SIF4, SIF5}.
We will resume the procedure as in the case presented by Gonçalves and Pinto about Wall
Street, namely:
a=2/3
I = BET-FI
n=5
s1 = SIF1
s2 = SIF2
s3 = SIF3
s4 = SIF4
s5 = SIF5.
First we will capture data from [13] (we choose 2nd semester of 2008) with a Visual Basic
for Application scripts. All the elements were also calculated necessary to assess the daily
fluctuation df. A chronogram of df can be view bellow:
data
12/2/08
12/16/08
11/18/08
11/4/08
10/7/08
10/21/08
9/23/08
9/9/08
8/26/08
8/12/08
7/29/08
7/1/08
1.0000
0.5000
0.0000
-0.5000
-1.0000
-1.5000
-2.0000
-2.5000
-3.0000
7/15/08
df(data)
2/3 df pentru BVB
df
From df series we build a new series z by:
zi = s - ln (- ln (dfi)) / b
where parameters b and s are specific of BHP distribution:
b = 0,936 ± 0,002
s = 0,374 ± 0,001
Testing universality of df are reduced at testing uniformity on (0,1) for z. A simple
histogram for df (see next figure) shows that we have not uniformity, agglomeration benefits
in an area value of 0.5:
Histogram
40
Frequency
30
20
10
Mean =0.53245
Std. Dev. =0.12008
N =125
0
0.20000
0.40000
0.60000
0.80000
1.00000
z
This can not be certified unless a statistical test is applied. Applying Kolmogorov-Smirnov
test for uniform distribution using SPSS 15.0 obtain a low degree of significance, which
convince us that the uniform distribution for z is rejected, i.e. the assumption of universality
is rejected for df:
One-Sample Kolmogorov-Smirnov Test
N
Uniform Parameters(a,b)
Most Extreme Differences
Kolmogorov-Smirnov Z
Minimum
Maximum
Absolute
Positive
Negative
z
125
.29356
.93461
.377
.377
-.151
4.216
Asymp. Sig. (2-tailed)
Monte Carlo Sig. (2-tailed)
Sig.
95% Confidence
Interval
Lower Bound
Upper Bound
.000
.000(c)
.000
.000
a Test distribution is Uniform.
b Calculated from data.
c Based on 10000 sampled tables with starting seed 299883525.
References
[1] Tiberiu Socaciu, Mirela Danubianu, BHP Universality Hypothesis Verification for
BET-FI Index of Bucharest Stock Exchange, in The Annals of the "Stefan cel Mare"
University Suceava. Fascicle of the Faculty of Economics and Public Administration,
vol. 8, no. 1(9), 2009, pp. 320-325.
[2] S. T. Bramwell, P. C. W. Holdsworth, J. F. Pinton, Universality of rare fluctuations
in turbulence and critical phenomena, in Nature, vol. 396, 1998, pp. 552-554.
[3] S.T. Bramwell, J.Y. Fortin, P.C.W. Holdsworth, S. Peysson, J.F. Pinton, B.
Portelli, M. Sellitto, Magnetic fluctuations in the classical XY model: The origin of an
exponential tail in a complex system, in Phys. Rev. E, 63, 041106, 2001
[4] C. Pennetta, E. Alfinito, L. Reggiani, S. Ruffo, Non-Gaussianity of Resistance
Fluctuations Near Electrical Breakdown, preprint online at
http://arxiv.org/PS_cache/cond-mat/pdf/0310/0310643v1.pdf, last access at 2009
january 20, last update at 2003 octomber 28.
[5] C. Pennetta, E. Alfinito, L. Reggiani, S. Ruffo, Non-Gaussian Resistance Noise near
Electrical Breakdown in Granular Materials, preprint online at
http://arxiv.org/PS_cache/cond-mat/pdf/0401/0401352v1.pdf, last access at 2009
january 5, last update at 2004 january 20.
[6] C. Pennetta, E. Alfinito, L. Reggiani, Resistor Network Approach to Electrical
Conduction and Breakdown Phenomena in Disordered Materials, online at
http://www.cmtg.it/leccemeeting/prin_networkw.ppt, last access at 2009 january 9.
[7] Carroll Croarkin, Paul Tobias, NIST/SEMATECH e-Handbook of Statistical
Methods, online at http://www.itl.nist.gov/div898/handbook/ , last access at 2009
january 10.
[8] Fisher-Tippett distribution, in The Free Dictionary by Farlex, online at
http://encyclopedia.thefreedictionary.com/Gumbel+distribution, last access at 2009
january 5.
[9] Tiberiu Socaciu, Ioan Maxim, PP Plot and QQ Plot tool for BHP universal
distribution, in AFASES 2009 - The International Session of XI-th Scientific Papers
"Scientific Research and Education in the Air Force", 20-22 MAY, BRASOV, 2009,
pp. 1162-1164, ISBN 978-973-8415-67-6.
[10] Probability-Probability (PP) Plot, Quantile-Quantile (QQ) Plot, online at
http://wiki.math.yorku.ca/images/0/08/Statistics_pp-qq-1.pdf, last access 2009 january
20.
[11] Tiberiu Socaciu, Mirela Danubianu, Ioan Maxim, Cornel Iosif, A Test of
Concordance for Nonparametric Bramwell-Holdsworth-Pinton Distribution, in The
33rd Annual Congress of the American Romanian Academy of Arts and Science (ARA),
Proceedings, volume II, Alma Mater University of Sibiu, June 02-07, 2009, pp. 309312, ISBN 978-2-553-01433-8.
[12] Rui Gonçalves, Albert Pinto, Universality in the stock exchange, preprint online at
http://arxiv.org/pdf/0810.2508v1, last update at 2008 october 14, last access 2009
january 20.
[13] http://www.bvb.ro